Page:An Investigation of the Laws of Thought (1854, Boole, investigationofl00boolrich).djvu/135

CHAP. VIII.] and in these expressions replace, for simplicity, we shall have from the three last equations,  and from this system we must eliminate $\scriptstyle{w}$. Multiplying the second of the above equations by $\scriptstyle{c}$, and the third by $\scriptstyle{c'}$, and adding the results to the first, we have When $$\scriptstyle{w}$$ is made equal to $\scriptstyle{1}$, and therefore $$\scriptstyle{\bar{w}}$$ to $\scriptstyle{0}$, the first member of the above equation becomes  And when in the same member $$\scriptstyle{w}$$ is made $$\scriptstyle{0}$$ and $\scriptstyle{\bar{w}=1}$, it becomes  Hence the result of the elimination of $$\scriptstyle{w}$$ may be expressed in the form  and from this equation $$\scriptstyle{x}$$ is to be determined. Were we now to proceed as in former instances, we should multiply together the factors in the first member of the above equation; but it may be well to show that such a course is not at all necessary. Let us develop the first member of (4) with reference to $\scriptstyle{x}$, the symbol whose expression is sought, we find  whence we find,  and developing the second member with respect to $$\scriptstyle{y}$$ and $\scriptstyle{z}$,